In the algebraic numerical theory, a part of mathematics, the Iwasawa theory is the study of objects of arithmetic importance over infinite numbers of towers of numbered bodies. The Iwasawa theory began as a Galois module theory of ideal class groups and was initiated in the 1950s by Japanese mathematician Kenkichi Iwasawa, as part of the theory of cyclotomy fields. In the early 1970s, Barry Mazur studied generalizations of the Iwasawa theory into abelian varieties. More recently (in the early 1990s), Ralph Greenberg has proposed an Iwasawa theory for motives.
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